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The largest class of multivalued systems satisfying the module-like axioms are the Hv -modules. The main tools concerning the class of Hv -modules with the ordinary modules are the fundamental relations. Based on the relation ε* , exact sequences in Hv -modules are defined. In this paper, we introduce the Hv -module M[A] and determine its heart and the connection between equivalence relations ε* M[A] and ε* A . Moreover, we define the M[-] and -[M] functors and investigate the exactness and some concepts related to them. Finally, we prove the five short lemma in Hv -modules. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that diagram lemmas for abelian groups (and more generally in abelian categories) used in algebraic topology, can be suitably extended to “non-abelian” structures such as groups, rings, loops, etc. Moreover, they are equivalent to properties which arise in the axiomatic study of these structures. For the five lemma this is well known, and in the present paper we establish this for the snake lemma and the lemma, which, when suitably formulated, turn out to be equivalent to each other for all (pointed) algebraic structures, and also in general categories of a special type. In particular, we show that among varieties of universal algebras, they are satisfied precisely in the so-called (pointed) ideal determined varieties. [Copyright &y& Elsevier]

Abstract: Fuhrmann’s state-space construction (in its generalized form) is used to obtain a general theory of first order representations of Fliess models defined over an arbitrary noetherian commutative ring. The case of arbitrary linear delay differential equations is involved. [Copyright &y& Elsevier]